Abstract

In Schrödinger picture we study the possible effects of trans-Planckian physics on the quantum evolution of massive nonminimally coupled scalar field in de Sitter space. For the nonlinear Corley-Jacobson type dispersion relations with quartic or sextic correction, we obtain the time evolution of the vacuum state wave functional during slow-roll inflation and calculate explicitly the corresponding expectation value of vacuum energy density. We find that the vacuum energy density is finite. For the usual dispersion parameter choice, the vacuum energy density for quartic correction to the dispersion relation is larger than for sextic correction, while for some other parameter choices, the vacuum energy density for quartic correction is smaller than for sextic correction. We also use the backreaction to constrain the magnitude of parameters in nonlinear dispersion relation and show how the cosmological constant depends on the parameters and the energy scale during the inflation at the grand unification phase transition.

1. Introduction

In the standard inflationary scenario, usual realization of inflation is associated with a slow rolling inflation minimally coupled to gravity [1]. Nevertheless, it is well known that the extension to the nonminimal coupling with the Ricci scalar curvature can soften the problem related to the small value of the self-coupling in the quartic potential of chaotic inflation [2]. Further, nonminimal coupling terms also can lead to corrections on power spectrum of primordial perturbations [3], a tiny tensor-to-scalar ratio [4, 5], and non-Gaussianities [6]. A broad class of models of chaotic inflation in supergravity with an arbitrary inflation potential was also proposed. In these models the inflation field is nonminimally coupled to gravity [7, 8]. Recently, the viability of simple nonminimally coupled inflationary models is assessed through observational constraints on the magnitude of the nonminimal coupling from the BICEP2 experiment [9].

Moreover, the standard inflationary scenario has two possible extensions. The first extension is associated with the ambiguity of initial quantum vacuum state, and the choice of initial vacuum state affects the predictions of inflation [10, 11]. The second extension concerns with the trans-Planckian problem [12, 13] of whether the predictions of standard cosmology are insensitive to the effects of trans-Planckian physics. In fact, nonlinear dispersion relations such as the Corley-Jacobson (CJ) type were used to mimic the trans-Planckian effects on cosmological perturbations [12–14]. These CJ type dispersion relations can be obtained naturally from quantum gravity models such as Horava gravity [15, 16]. Recently, in several approaches to quantum gravity, the phenomenon of running spectral dimension of spacetime from the standard value of 4 in the infrared to a smaller value in the ultraviolet is associated with modified dispersion relations, which also include the CJ type dispersion relations [17, 18].

In the previous work [19–23] we used the lattice Schrödinger picture to study the free scalar field theory in de Sitter space, derived the wave functionals for the Bunch-Davies (BD) vacuum state and its excited states, and found the trans-Planckian effects on the quantum evolution of massless minimally coupled scalar field for the CJ type dispersion relations with sextic correction. In this paper we extend the study to the case of massive nonminimally coupled scalar field.

The paper is organized as follows. In Section 2, the theory of a generically coupled scalar field in de Sitter space is briefly reviewed in the lattice Schrödinger picture. In Section 3, we consider the massive nonminimally coupled scalar field during slow-roll inflation and use the CJ type dispersion relations with quartic or sextic correction to obtain the time evolution of the vacuum state wave functional. In Section 4, using the results of Section 3, we calculate the finite vacuum energy density, use the backreaction to constraint the parameters in nonlinear dispersion, and evaluate the cosmological constant. Finally, conclusions and discussion are presented in Section 5. Throughout this paper we will set .

2. De Sitter Scalar Field Theory in Schrödinger Picture

In this section, we begin by briefly reviewing the theory of a generically coupled scalar field in de Sitter space in the lattice Schrödinger picture (for the details of some derivations in this section see [23]). The Lagrangian density for the scalar field we consider is where is a real scalar field, is the potential, is the mass of the scalar quanta, is the Ricci scalar curvature, is the coupling parameter, and = , . For a spatially flat ()-dimensional Robertson-Walker spacetime with scale factor , we have In the ()-dimensional de Sitter space we have , where is the Hubble parameter which is a constant.

For , in the lattice Schrödinger picture, we obtain from (2) the time-dependent functional Schrödinger equation in momentum space [23] where Here , , that is, is the overall comoving spatial size of lattice, , , is the conjugate momentum for , and the subscripts 1 and 2 denote the real and imaginary parts, respectively.

For each real mode , we have Note that (8) arises from the field quantization of the Hamiltonian (5) through the functional Schrödinger representation , , where operators and satisfy the equal time commutation relations , and setting so that . Thus (8) governs the time evolution of the state wave functional of the Hamiltonian operator in the representation. In terms of the conformal time defined by the normalized wave functionals of vacuum and its excited states are with the amplitude and phase Here is defined by , is the th-order Hermite polynomial, is the Hankel function of the first kind of order , , and the prime in (12) denotes the derivative with respect to . The complete wave functionals can be written as , where means that mode is in the excited state, mode is in the excited state, and so forth. For , the ground state wave functional corresponds to the BD vacuum.

For , we have , , and the mode index in carries labels which will be suppressed below. Furthermore, from (3)–(8) we get in the continuum limit ()

3. Trans-Planckian Effects on Vacuum Wave Functional

For the inflationary potential , the bounds on derived from the joint data analysis of Planck + WP + BAO + high- for the number of -foldings are (68% CL), (95% CL) [24].

For the mass in the tree-level potential, we have GeV [25]. Moreover, the recent BICEP2 experiment suggests that GeV [26–28]. Therefore, from , we have and , which will be used below. To study further the effects of trans-Planckian physics, we use the CJ type dispersion relation where is a cutoff scale, is an integer, and is an arbitrary coefficient [12–14].

3.1. CJ Type Dispersion Relations with Quartic Correction

First, we use the CJ type dispersion relation (14) with and to obtain the time evolution of the vacuum state wave functional. Recall that these CJ type dispersion relations can be obtained from theories based on quantum gravity models [15–18].

Using which is the ratio of physical wave number to the inverse of Hubble radius, (13) becomes where , and the ground state solution of (15) becomes where and satisfy In region I where , that is, , the dispersion relations can be approximated by , and the corresponding wave functional for the initial BD vacuum state is [23, 29] where the prime in (20) denotes the derivative with respect to .

On the other hand, in region II where , that is, , linear relations recover , and the corresponding wave functional for the non-BD vacuum state is [23, 29] where the prime in (22) denotes the derivative with respect to and and satisfy . Let be the time when the modified dispersion relations take the standard linear form. Then where for . The constants and can be obtained by the following matching conditions at for the two wave functionals (19) and (21): which can also be rewritten, respectively, as by requiring , , and when .

Using with and , we have from (20), (22), and (25) where we choose and , and is a relative phase parameter. Then from (27) and we have where , . Substituting (20) and (22) into (26) and keeping terms up to order on the right-hand side of (26), we obtain Using (28) in (29) gives Here we choose , so that is small for to avoid an unacceptably large backreaction on the background geometry. Then we have or

3.2. CJ Type Dispersion Relations with Sextic Correction

In this subsection, we use the CJ type dispersion relation (14) with and to obtain the time evolution of the vacuum state wave functional. For this case, only (15), (18), and (20) are changed into where , and the prime in (35) denotes the derivative with respect to . Using with and for , we obtain from (35), (22), (25), and where , . Substituting (35) and (22) into (26) and keeping terms up to order on the right-hand side of (26), we find Using (37) in (38) gives Here we choose , so that is small for to avoid an unacceptably large backreaction on the background geometry. Then we have or

4. Vacuum Energy, Backreaction, and Cosmological Constant

Using the results of Section 3, we proceed to calculate the finite vacuum energy density and use the backreaction constraint to address the cosmological constant problem. Note that, in the slow-roll approximation, the energy density of the scalar field is , where . Therefore the relation between the expectation value of the vacuum energy density and the vacuum wave functional in (16) is where we use a field redefinition , denotes the real part of , and the factor in (43) appears through the normalization condition